Piezoelectric vibrator and method for manufacturing the same

ABSTRACT

In a piezoelectric vibrator which has excitation electrodes respectively disposed on the top surface and undersurface of a piezoelectric substrate, at least one of the excitation electrodes being an inverted-mesa electrode that has a recess formed in the opposite side of the electrode from the side which contacts the piezoelectric substrate. Thus, an AT-cut quartz resonator in which the capacitance ratio γ is reduced to a value below the limit value can be provided as a result of use of the inverted-mesa electrodes as excitation electrodes.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention relates to a piezoelectric vibrator which is usedin, for instance, frequency voltage-controlled oscillators (VCOs) andmore particularly to an AT-cut quartz resonators in which thecapacitance ratio γ is reduced by the use of an inverted-mesa electrode.

2. Description of Prior Art

Generally, quartz crystal resonators have a high Q value, so that thefrequency stability of such resonator is high. The resonators are widelyused in radio apparatus as reference oscillators, etc. AT-cut quartzresonators are particularly superior in terms of temperature-frequencycharacteristics.

As shown in FIG. 22, in a conventional an AT-cut quartz resonator,disk-form electrodes 3a and 3b, which are smaller than the diameter of acircular AT-cut quartz crystal substrate 1, are provided on both sidesof the central portion 2 of the substrate 1, and lead wires forsupplying voltage are led out from these electrodes 3a and 3b. Here, aparallel-plane substrate or a substrate which has been subjected tobevel working, etc. is used for the substrate 1; and gold, silver,aluminum, nickel or alloys of these metals, etc. are used as theelectrodes 3a and 3b.

The AT-cut quartz resonator described above is generally used in afrequency voltage-controlled oscillator (VCO) in which the oscillationfrequency is controlled by a voltage from the outside, thus constitutinga frequency voltage-controlled quartz crystal oscillator (VCXO). TheAT-cut quartz resonators used in such applications should thereforeoffer not only a broad adjustable range of frequencies but also anexcellent frequency stability when the control voltage is constant.

However, the VCXO that uses quartz resonators has problems: although thefrequency stability is high due to the fact that the Q value is higherthan that of other piezoelectric vibrators, the frequency adjustablerange is narrow, thus making it difficult to satisfy the above-describedrequirements.

The AT-cut quartz resonators, that is, vibrators formed by providingelectrodes of a uniform thickness to an AT-cut quartz substrate whichhas a parallel plane shape or which has been subjected to bevel working(as shown in FIG. 22) are used in such VCXOs, the frequency adjustablerange is narrow since the characteristics capacitance ratio γ of theAT-cut quartz resonator has a minimum limit value of approximately 200and cannot be lowered any further than such a value.

When the capacitance ratio γ of such an AT-cut quartz resonator isreduced, there are such merits as the short-term stability is improved,the frequency adjustable range is broadened when a filter constructionis employed, impedance can be lowered and excitation is facilitated inaddition that the frequency adjustable ranges of the VCO is broadened.Nevertheless, it has been considered impossible in the past to lower thecapacitance ratio γ of the AT-cut quartz crystal as oscillator describedabove to a value of 200 or lower.

The present invention is devised in light of the above facts; and theobject of the present invention is to provide an AT-cut quartz resonatorwhich makes it possible to reduce the capacitance ratio γ to a valuesmaller than the conventional characteristic capacitance ratio γ, thathas been recognized by a person skilled in the art, by usinginverted-mesa electrodes.

DISCLOSURE OF THE INVENTION

In order to accomplish the object, the present invention ischaracterized in that in a piezoelectric vibrator, in which excitationelectrodes are respectively disposed on the top surface and undersurfaceof a piezoelectric substrate, at least one of the excitation electrodesis an inverted-mesa electrode which has a recess formed in the oppositeside of the electrode from the side that contacts the piezoelectricsubstrate.

Another characteristic feature of the present invention is that theratio of mesa thickness and ratio of mesa length of the inverted-mesaelectrode are set so that the capacitance ratio of the piezoelectricvibrator is less than 200.

Another characteristic feature of the present invention is that thepiezoelectric substrate is an AT-cut quartz crystal substrate.

Another characteristic feature of the present invention is that theportion of the piezoelectric substrate that corresponds to a peripheralportion of the electrode located in the central portion of at least oneof the two sides of the piezoelectric substrate is formed with a greaterthickness than other portions of the substrate, and an excitationelectrode that has a uniform thickness is disposed on thiscentral-portion, thus forming the inverted-mesa electrode.

Another characteristic feature of the present invention is that therecess of the inverted-mesa electrode is formed in the shape of astairway.

Another characteristic feature of the present invention is that therecess of the inverted-mesa electrode is curved in a concave shape.

Another characteristic feature of the present invention is that therecess of the inverted-mesa electrode is curved in a convex shape.

Still another characteristic feature of the present invention is that ina method for manufacturing an AT-cut quartz resonator in whichexcitation electrodes are respectively disposed on the top surface andundersurface of a piezoelectric substrate, the method comprises a stepin which a piezoelectric substrate is prepared, a step in whichplate-form excitation electrodes are provided to both sides of thepiezoelectric substrate, and a step in which the central portion of thesurface of at least one of the excitation electrodes, which is on theopposite side of the electrode from the side in contact with thepiezoelectric substrate, is removed so as to form a recess.

Another characteristic feature of the present invention is that in amethod for manufacturing an AT-cut quartz resonator in which excitationelectrodes are respectively disposed on the top surface and undersurfaceof a piezoelectric substrate, the method comprises a step in which apiezoelectric substrate which is somewhat thick is prepared, a step inwhich a part of the piezoelectric substrate is removed from the centralportion of the surface on at least one of the two sides of thepiezoelectric substrate so that the portion of the substratecorresponding to the peripheral portion of the inverted-mesa electrodebecomes thicker than the other portions of the substrate, and a step inwhich an excitation electrode with a uniform thickness is disposed onthe central portion of the piezoelectric substrate so as to form aninverted-mesa electrode.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 illustrates the constitution of one embodiment of the AT-cutquartz resonator according to the present invention wherein FIG. 1(a) isa perspective view thereof, and FIG. 1(b) is a cross sectional viewtaken along the center line of the quartz crystal substrate shown inFIG. 1(a).

FIG. 2 is a diagram which illustrates an analytical model used toanalyze the principle of the present invention.

FIG. 3 is a frequency spectrum for the mode S₀ which is the lowestsymmetry inharmonic mode in AT-cut quartz resonator of present inventionand conventional AT-cut quartz resonator.

FIG. 4 is a graph which shows the conditions under which region I in themodel shown in FIG. 2 is flat.

FIG. 5 is a graph which shows the energy trapping factor of the point atwhich generation of the S₁ mode is initiated in the model shown in FIG.2.

FIG. 6 is a graph which shows the resonance frequency spectrum in themodel shown in FIG. 2.

FIG. 7 is a graph which shows an example of calculation of thedisplacement distribution in the model shown in FIG. 2.

FIG. 8 is a diagram which shows the equivalent circuit of the quartzresonator.

FIG. 9 is a graph which shows the relationship between the ratio of mesathickness μ and the capacitance ratio γ in the model shown in FIG. 2.

FIG. 10 shows the material constants of the AT-cut quartz resonator.

FIG. 11 is a diagram of the manufacturing process used to manufacture anAT-cut quartz resonator shown in FIG. 1(a), (b).

FIG. 12 illustrates the constitution of a second embodiment of theAT-cut quartz resonator of the present invention.

FIG. 13 is a diagram of the manufacturing process used to manufacturethe AT-cut quartz resonator shown in FIG. 12.

FIG. 14 illustrates a modified example in which only one of theelectrodes in the embodiment shown in FIG. 1 is formed with aninverted-mesa shape.

FIG. 15 illustrates a modified example in which only one of theelectrodes in the embodiment shown in FIG. 12 is formed with aninverted-mesa shape.

FIG. 16 illustrates a modified example in which the recess of theinverted-mesa excitation electrode is formed in the shape of a stairway.

FIG. 17 illustrates a modified example in which the recess of theinverted-mesa excitation electrode is curved in a concave shape.

FIG. 18 illustrates a modified example in which the recess of theinverted-mesa excitation electrode is curved in a convex shape.

FIG. 19 illustrates a modified example in which only one of theelectrodes in the embodiment shown in FIG. 16 is formed with aninverted-mesa shape.

FIG. 20 illustrates a modified example in which only one of theelectrodes in the embodiment shown in FIG. 17 is formed with aninverted-mesa shape.

FIG. 21 illustrates a modified example in which only one of theelectrodes in the embodiment shown in FIG. 18 is formed with aninverted-mesa shape.

FIG. 22 illustrates a conventional AT-cut quartz resonator in which FIG.22 (a) is a perspective view thereof, and FIG. 22 (b) is a sectionalview taken along the center line of the quartz crystal substrate shownin FIG. 22 (a).

PREFERRED EMBODIMENTS OF THE INVENTION

Below, the present invention will be described based upon theembodiments illustrated in the drawings. FIG. 1 illustrates thestructure of one embodiment of the AT-cut quartz resonator according tothe present invention wherein FIG. 1 (a) is a perspective view and FIG.1(b) is a sectional view taken along the centerline of the quartzcrystal substrate shown in FIG. 1(a).

As shown in FIG. 1, in this AT-cut quartz resonator, excitationelectrodes 7a and 7b which are smaller in diameter than a circularparallel flat-plate type AT-cut quartz substrate 1 are disposed on bothsides of roughly the central portion 2 of the AT-cut quartz substrate.As shown in FIG. 1(b), the excitation electrodes 7a and 7b are formed aninverted-mesa shape respectively in which the sides that contact thequartz crystal substrate have a flat planar shape and adheres closely tothe quartz crystal substrate 1. In addition, recesses 9a and 9b areformed in the central portions of the surfaces on the opposite sidesfrom the sides that contact the quartz crystal substrate. Morespecifically, as shown in FIG. 1(b), the thickness T1 of the centralportions 11a and 11b of the electrodes 7a and 7b is smaller by aprescribed amount than the thickness T2 of the peripheral portions 13aand 13b that surround the central portions 11a and 11b.

The feature of the present invention is that by forming the excitationelectrodes into inverted-mesa electrodes as described in the embodimentabove, the characteristic capacitance ratio γ is reduced to a valuelower than the limit value (approximately 200), which is conventionallyrecognized by persons skilled in the art as the value of quartzresonators that is formed by electrodes of a uniform thickness on anAT-cut quartz crystal substrate that has a parallel plane shape or uponwhich a bevel working has been performed.

In other words, the capacitance ratio γ can be reduced by formingelectrodes so as to have an inverted-mesa structure and not a uniformthickness structure which was previously thought to be appropriate.Accordingly, the relationship between the inverted-mesa shape of theexcitation electrodes and the capacitance ratio γ of the quartzresonator that has such excitation electrodes was examined using energytrapping analysis procedure of an isotropic plate, and the resultsobtained will be described below in detail.

The energy trapping displacement and distribution of AT-cut quartzresonator having a conventional electrode structure shows a partialcosine shape in the electrode portions and an exponential shape whichcontinuously connects the displacement and force, in the peripheralportions. A displacement shape was examined for the structure in whichthe electrode portions are of an inverted-mesa double electrodestructure. As a result, it was found that by varying the mesa shape, thedisplacement shape solution consisting of "central portion flatness"exists in a state in which no inharmonic spurious S1 mode is generated.

It was further found that when the capacitance ratio is observed whilevarying the mesa shape in this state, the capacitance ratio reaches aminimum value in the vicinity of this "central portion flatness"displacement shape. Moreover, the calculated value of this minimum valuewas 169.

Next, the analysis model used for analyzing the displacementdistribution of the inverted-mesa shape will be described:

The X-direction anisotropy of an AT-cut quartz resonator with respect tothe plate thickness and the Z-direction anisotropy of the oscillatorwith respect to the plate thickness are substituted for isotropicanalyses by the multiplication of correction coefficients. Accordingly,an analysis model that comprises an isotropic substrate 19 which hasinverted-mesa electrodes 17a and 17b and has the cross section as shownin FIG. 2 is envisioned.

As seen from the cross-sectional shape shown in FIG. 2, a quartzresonator with a structure in which electrodes 17a and 17b having alength of 2a are disposed on the surfaces of a substrate 19 that has aplate thickness of H is envisioned. The vibrational mode is an SH wave.

The portions of the electrodes 17a and 17b extending from point p topoint a have a greater thickness than the other portions of theelectrodes, so that the electrodes as a whole have an inverted-mesashape.

Here, the displacement U is assumed to be as shown below; and it is alsoassumed that the displacement U is uniformly distributed in the Xdirection. ##EQU1##

Here, U_(i) is a function of Z and is given for the respective regionsI, II and III in FIG. 2 by the following equations using the arbitraryconstants A, B, C, D and E:

    U.sub.i1 =A cos k.sub.1 Z                                  (2)

    U.sub.i2 =B cos k.sub.2 Z+C sin k.sub.2 Z                  (3)

    U.sub.i3 =D cosh k.sub.3 Z+E sinh k.sub.3 Z                (4)

Here, the wave numbers k₁, k₂ and k₃ are the propagation constants forthe respective regions and are given by the following equations:##EQU2## wherein, f is the resonance frequency, f₁, f₂ and f₃ are thecut-off frequencies in the respective regions in FIG. 2, t indicatestime, and n is the order of the overtone.

If the ratios of the arbitrary constants A, B, C, D and E are determinedunder the conditions that there is no discontinuity in displacement andstress when Z=p and Z=a in FIG. 2, and that no stress when Z =b, thenthe following frequency equation is obtained: ##EQU3## Here,

    P=tan k.sub.2 (a-p)                                        (8a) ##EQU4##

To perform a numerical calculation on Equation (8) by substitutingdimensional data, the following definitions are given. ##EQU5##

Here, Δ means the mass loading effect of region I with respect to regionIII, ψ is the normalized resonance resonance frequency, and f is theresonance frequency.

Furthermore, the following quantity is defined as the energy trappingfactor: ##EQU6##

Furthermore, the ratio of mesa-thickness (μ) is defined as follows, andthe parameter (p/a) is defined as the ratio of mesa-length. ##EQU7##

Next, the frequency spectrum for the mode S₀ in a case where theabove-described analytical model is used will be described. First, anumerical calculation was performed for the lowest symmetry inharmonicmode S₁ using the frequency equation in Equation (8), and the resultsobtained are shown in FIG. 3. The vertical axis is the normalizedresonance frequency ψ, ψ=1 corresponds to the cut-off frequency inregion III, and ψ=0 corresponds to the cut-off frequency in region I.Furthermore, the horizontal axis is the energy trapping factor. Aspecial feature of the numerical calculation results shown in the samefigure is that ψ is zero when the energy trapping factor isapproximately 0.6, as indicated by the solid line. Furthermore, forreference, the numerical calculation results obtained in a case wherethe thickness of the portion corresponding to region II is made the sameas that of the portion corresponding to region I are indicated by adotted line. This corresponds to a common simple electrode structurewhich is not an inverted-mesa double electrode structure. As is clearfrom FIG. 3, ψ never reaches zero.

More specifically, as is clear from the numerical calculation resultsshown in FIG. 3, it was ascertained that the use of an inverted-mesashape for the electrode structure results in the existence of acombination of structural parameters which is such that the normalizedresonance frequency ψ is equal to zero (i. e., which is such that theresonance frequency f and the cut-off frequency f₁ are equal). Thesignificance of this normalized resonance frequency being equal to zerowill be discussed in detail below; however, this means that the wavenumber k1 of the central portions of the electrodes (region I) is zero,i. e., that the displacement U_(i1) of this region is flat. In thiscase, the capacitance ratio is at a minimum.

Next, whether or not structural parameters which are such that thenormalized resonance frequency ψ is always zero exist in a case where aninverted-mesa double electrode structure is used as the electrodestructure of the quartz resonator will be examined.

Equation (8) can also be considered as a function of both the normalizedresonance frequency in Equation (10) and the energy trapping factor inEquation (11), and the differential coefficient is determined, then itis found as follows:

1) When the ratio of mesa-thickness μ is zero, the larger the value ofthe abscissa (ψ=0), the smaller is the absolute value of the firstderivative. The locus of the function approaches the abscissaasymptotically. The dotted line in FIG. 3 represents this case with aninfinite plate for reference.

2) When the ratio of mesa-thickness μ is positive, then as the value ofthe abscissa increases, the absolute value of the first derivative doesnot decrease and is almost constant. Hence, it is natural to assume thatthe locus will definitely cross the abscissa(ψ=0).

Under conditions which are such that the normalized resonance frequencyψ is equal to zero, i. e., when the resonance frequency f and thecut-off frequency f₁ of region I are equal, the wave number k₁ of regionI from Equation (5) becomes zero as shown by the equation below. As isalso clear from Equation (2), this is a state in which the displacementU_(i1) of region I is flat.

    k.sub.1 =0                                                 (13)

Next, we will examine combinations of structural parameters which aresuch that the displacement of the central portion of the electrode(region I) is flat, i.e., which are such that k1=0. Generally, theconditions under which the central portion of the electrode is flat aregiven by the following equation:

    k.sub.3 tanh k.sub.3 (b-a)=k.sub.2 tan k.sub.2 (a-p)       (14)

FIG. 4 shows the results obtained when calculations were performed bysubstituting Equations (6) and (7) for k₂ and k₃ in the above-describedEquation (14), extracting only the structural parameters necessary fordesign, and varying the ratio of mesa thickness μ. The horizontal axisindicates a quantity obtained by multiplying the energy trapping factorby the substrate dimension taken with the electrode dimension as areference {(b-a)/a}, and the vertical axis indicates a quantity obtainedby multiplying the energy trapping factor, the ratio of the mesa length(1 -(p/a)) and the ratio of the mesa thickness μ. Furthermore, thevalues on both the vertical axis and the horizontal axis are abstractnumbers, here, the vertical axis is a function of the electrode recessdimension p, and the horizontal axis is a function of the substratedimension b.

It is seen from FIG. 4 that the value on the vertical axis eventuallyreaches a fixed value as the value on the horizontal axis increases.This is inferred to mean that as the difference (b-a) between thedistance b from the center of the quartz crystal substrate to the endportion of the substrate and the distance a from the center of thequartz crystal substrate to the end portion of the electrode increases,i.e., as region III increases in size, the effects of the substrate endportions on structural parameters such as the dimensions of region I andregion II and the thickness, etc., are eliminated. Furthermore, even ifthe value on the horizontal axis is small, the value on the verticalaxis has some magnitude; and it is seen from this that even if thedimensions of region III are small, structural parameters exist whichare such that the wave number k₁ is equal to zero. In other words, byinstalling inverted-mesa electrodes, it is possible to obtain a wavenumber k₁ that is equal to zero even if a small-diameter quartz crystalsubstrate is used.

The setting of structural parameters will be described further.

In a quartz resonator, serious problems occur when an unwanted responseof any inharmonic mode appears in the vicinity of the resonancefrequency. Here, therefore, the energy trapping factor is chosen so thatthe lowest-symmetry inharmonic mode S₁ is observable.

FIG. 5 shows the calculation of the point at which the generation of thelowest-symmetry inharmonic mode S₁ is observable in a quartz resonatorwhich has an inverted-mesa electrode shape. Here, the vertical axis isthe energy trapping factor, the horizontal axis is the ratio of mesalength (1-(p/a)), and the parameter is the ratio of mesa thickness μ.For example, in the case of the analytical model used for the quartzresonator with an inverted-mesa double electrode structure shown in FIG.2, if the structural parameters of the quartz resonator are set so thatthe ratio of mesa length (p/a)=0.5 and the ratio of mesa thicknessμ=0.5, then the energy trapping factor at which the S₁ mode is generatedbecomes 0.6 or higher. Meanwhile, as shown by the numerical results inFIG. 3, the energy trapping factor, at which the normalized resonancefrequency ψ=0, is 0.6. Thus, it can be seen that if the above-describedstructural parameters and parameters concerning the energy trappingfactor are set so that the energy trapping factor at which the S₁ modeis generated is larger than the energy trapping factor at which ψ=0,then the inharmonic spurious caused by the S₁ mode will not begenerated.

Next, the question of what kind of behavior is shown by a quartzresonator that has an inverted-mesa double electrode structure in a casewhere the relationship between the energy trapping factor and thenormalized resonance frequency is such that ψ=0 or less will beexamined. In the frequency equation that applies in this case, thepropagation constant for region I is an imaginary number; accordingly,it is necessary to perform a numerical analysis by replacing thisconstant with a real number and then determining a new frequencyequation. The results of such a numerical analysis are shown in FIG. 6.In the area of FIG. 6 in which the vertical axis ψ is above zero, thedisplacement distribution of region I shows a cos shape; and in the areain which the vertical axis ψ is below zero, this distribution shows acosh shape. Furthermore, as in the case of FIG. 3, ψ=1 is the cut-offfrequency for region III, and ψ=0 is the cut-off frequency for region I.Moreover, in this example of calculation, the ratio of mesa thickness μis 0.5, the ratio of mesa length (p/a) is 0.5, and the ratio of thesubstrate diameter to the electrode diameter (b/a) is 11. In addition,the position where the normalized resonance frequency ψ=-0.5 is theposition of the cut-off frequency for region II.

It is clearly seen from FIG. 6 that the value varies continuously aboveand below the horizontal axis where ψ=0.

In a case where the displacement distribution of region I shows a cosshape, the displacements of regions I, II and III are respectively givenby the following equations:

    U.sub.i1 =A cos k.sub.1 Z                                  (15) ##EQU8##

Here, r can be expressed by the following equation using P and Q inEquations (8a) and (8b). ##EQU9##

In a case where the displacement distribution of region I shows a coshshape, the displacements can likewise be determined using Equations(15), (16) and (17) by replacing the propagation constant of region Iwith a real number.

FIG. 7 shows an example of calculation of displacement distributions. Asshown in this figure, it was found that the vibrational displacementshape can be greatly varied under conditions which produce no inharmonicspurious by using electrodes that have an inverted-mesa structure andappropriately selecting the energy trapping factor. The displacementdistribution in the case of a ratio of mesa thickness μ is 0.5 is adisplacement shape with a flat distribution in the central portion. Incases where the ratio of mesa thickness μ is less than 0.5, asingle-peak distribution is obtained; and in cases where the ratio ofmesa thickness μ is greater than 0.5, a double-peak distribution isobtained.

Next, how the capacitance ratio, which is the ratio of the clampedcapacitance C₀ in FIG. 8 to the motional capacitance C₁ of a seriesresonance circuit, C₀ /C₁, varies in a case where the shape of thevibrational displacement is greatly varied by using electrodes with aninverted-mesa structure and appropriately selecting the energy trappingfactor will be determined.

First, the calculation of the equivalent inductance L of the resonatoris one of the steps in determining the capacitance ratio. In thiscalculation, it should be assumed that the time average of theelectromagnetic energy is equal to that of the kinetic energy, and thatthe electro-magnetic energy is stored in this equivalent inductance. Inthis case, the following approximations should be used:

1. As a quartz plate is a low-coupling piezoelectric material, thepropagation constant in the thickness direction (y' axis) is given bythe product of an odd number and π/2.

2. The electric field is uniform in the X-Z' plane, and thus has nodependence on either the X or Y' axis.

Under these approximate conditions, the capacitance ratio is given bythe following equation: ##EQU10##

Here, k₂₆ is the electromechanical coupling factor, n is the order ofovertone in the direction of thickness; the displacement consists of theZ-direction and X-direction dependent portions of the displacement inEquation (1); and as a measure of the dimensions, the electrodedimensions in the respective directions are used as a standard.Furthermore, area integration of the denominator is performed only forthe electrode surface (Se), and area integration of the numerator isperformed for the vibrator as a whole (Sw).

If the mesa shape changes, the displacement shape also changesaccordingly, and the capacitance ratio changes with the change in thedisplacement shape. Independent parameters which regulate thedisplacement distribution include the ratio of mesa thickness μ and theratio of mesa length (p/a). The capacitance ratio of a quartz resonatorwith an inverted-mesa electrode structure having the structuralparameters shown in FIG. 7 can be determined using Equation (18), withthe above-described ratio of mesa thickness μ and ratio of mesa length(p/a) given. Furthermore, if the energy trapping factor at which theinharmonic spurious S₁ begins to be generated is selected using FIG. 5,then all of the parameters are determined. More specifically, the ratioof mesa length (p/a) is set at 0.5, and five points, i.e., 0.1, 0.3,0.5, 1 and 2 are selected for the ratio of mesa thickness μ. Thedisplacements in the case of these parameter settings are expressed byEquations (15), (16) and (17). FIG. 9 shows the results obtained whennumerical calculations were performed by substituting these values intothe capacitance ratio equation of Equation (18). The horizontal axis isthe ratio of mesa thickness μ, and the vertical axis is the capacitanceratio.

In a case where the ratio of mesa thickness μ=0.5, the capacitance ratiotakes a minimum value. Here, the electro-mechanical coupling coefficientof an AT quartz crystal plate is approximately 8.9%. This equals acapacitance ratio of 126, which is determined by the material constants.Since the displacement distribution actually has the form of a sine(sin) function in the direction of thickness, this increases to 156.Furthermore, this is increased by the shape effect. Here, the values inthe table shown in FIG. 10 were used to determine the couplingcoefficient.

When the ratio of mesa thickness μ is 0, this corresponds to the case ofa conventional simple electrode rather than an inverted-mesa shape. Thecapacitance ratio in this case is approximately 171, while thecapacitance ratio in the case of a ratio of mesa thickness μ is 0.5 is169, so that the capacitance ratio is decreased by approximately 2%.Furthermore, when the displacement distribution has a twin peaks shapeor when the ratio of mesa thickness μ exceeds 0.5 or more, then it isseen that the capacitance ratio increases greatly as shown in FIG. 9.Moreover, the use of an inverted-mesa electrode structure results in adecrease of approximately 2% in the capacitance ratio compared to thecase of simple electrodes, and this is because the unit length in the Xdirection is considered to be an infinite length. In actuality, thedisplacement distribution also varies in this X direction; andtherefore, the degree of variation in the capacitance ratio increases,and the effect of the inverted-mesa is enhanced.

Thus, the following results were obtained by way of use of an isotropicone-dimensional analytical method for the energy confinementcharacteristics of an AT-cut quartz resonator with an inverted-mesaelectrode structure:

With the energy trapping factor-at which the lowest-order inharmonicspurious (S1) begins to be generated selected as a condition ofanalysis,

1. The displacement distribution becomes flat at the central region ofthe electrode depending on the shape and dimensions of the mesa (i.e.,ratio of mesa thickness and ratio of mesa length) and the energytrapping factor. Two displacement distributions with a single peak andtwin peaks are also observed in accordance with the choice ofparameters.

2. The conditions under which the displacement distribution becomes flatin the central region of the electrode were given.

3. The capacitance ratio becomes minimum where the displacementdistribution is flat.

4. The minimum capacitance ratio of an AT-cut quartz resonator with theinverted mesa electrode has a minimum value 169.

It is seen from these facts that the capacitance ratio γ of theabove-described quartz resonator can be lowered to a value below thelimit value (approximately 200), i.e., to a value of 169 in the presentembodiment, because of the use of an inverted-mesa shape for theexcitation electrodes of the quartz resonator.

Furthermore, in the above Equation (12), the ratio of mesa thickness μis expressed as ##EQU11## in terms of the cut-off frequencies; here,since the cut-off frequency f can generally be expressed as ##EQU12##(k: constant determined by material, t: plate thickness) as a functionof the plate thickness t, it can be said that μ expresses the thicknessratio.

Next, the method used to manufacture an AT-cut quartz resonator shown inthe FIG. 1 will be described.

FIG. 11 is a diagram of the manufacturing process for manufacturing anAT-cut quartz resonator shown in FIG. 1. First, as shown in FIG. 11(a),a circular AT-cut quartz crystal substrate 1 is prepared; then, as shownin FIG. 11(b) small-diameter excitation electrodes 7a and 7b areprovided to both surfaces of roughly the central portion 2 of the AT-cutquartz crystal substrate 1; and then as shown in FIG. 11(c), the centralportions of the electrodes 7a and 7b are removed by a method such asetching, etc. so as to form recesses 9a and 9b, thus producing aninverted-mesa shape.

FIG. 12 is a structural diagram which illustrates a second embodiment ofthe AT-cut quartz resonator of the present invention. As shown in FIG.12, this second embodiment has a structure in which the portions 23a and23b of a circular AT-cut quartz crystal substrate 21 that correspond tothe peripheral portions of the electrodes located in the centralportions on both sides are formed so as to have a greater thickness thanthe other portions of the substrate 21, and excitation electrodes 25aand 25b with a uniform thickness are formed on top of these portions.Thus, the above-described electrodes 25a and 25 have an inverted-mesashape with respective recesses 27a and 27b. The function and effect ofthis second embodiment are the same as those of the above-describedfirst embodiment.

The method used to manufacture the above-described second embodimentwill be described with reference to FIG. 13. First, as shown in FIG.13(a), a circular AT-cut quartz crystal substrate 29 which is slightlyon the thick side is prepared. Then, as shown in FIG. 13(b), this quartzcrystal substrate 29 is etched so that portions 23a and 23b whichcorrespond to the peripheral portions of the electrodes 25a and 25b areleft; and as shown in FIG. 13(c), excitation electrodes 25a and 25bwhich have a uniform thickness are formed on both surfaces of thecentral portion of the quartz crystal substrate 29, so that theelectrodes 25a and 25b have an inverted-mesa shape.

Next, examples of measurements obtained for the embodiment shown in FIG.12 will be described.

Here, in the quartz resonator shown in FIG. 12, vacuum-evaporated filmsof Cr-CrAu-Au (total film thickness 3000Å (angstroms)) with a diameter Cof 3.2 mm were caused to adhere to the central portions of an AT-cutcircular polished plate 21 that has a diameter A of 7.0 mm, and etchingwas performed with ammonium fluoride at 60° C., so that portions otherthan the above-described portions 23a and 23b were removed on both sidesby a total amount 2D of 2.3 μm (D on one side =1.15 μm). Then, Au filmswith a thickness E of 2900Å (angstroms) were applied on top of theseportions by vacuum evaporation, thus forming excitation electrodes 25aand 25b. When this resonator was placed in an HC-18/U holder and theequivalent constants were measured, the following results were obtained:

Frequency: 10.698760 MHz

Equivalent resistance: 20 Ω

Equivalent series capacitance(mo: 14.85 PF

Parallel capacitance: 2.60 PF

Capacitance ratio: 175

Incidentally, when a sample with ordinary excitation electrodes withoutan inverted-mesa shape was measured under the above-describedconditions, the capacitance ratio was 210.

The quartz resonator with inverted-mesa electrodes provided by thepresent invention is not limited to those described in the first andsecond embodiments, and various modifications such as those describedbelow are possible.

More specifically, FIG. 14 shows an example in which only one electrode7a has an inverted-mesa shape in the embodiment shown in FIG. 1; andFIG. 15 shows an example in which only the electrode 25a on one side inthe embodiment shown in FIG. 12 has an inverted-mesa shape.

Moreover, FIG. 16 shows an example in which the recesses 32a and 32b ofthe inverted-mesa excitation electrodes 31a and 31b have a "stairway"shape, and FIG. 17 shows an example in which the recesses 34a and 35b ofthe inverted-mesa excitation electrodes 33a and 33b are curved in aconcave shape. Furthermore, FIG. 18 shows an example in which therecesses 36a and 36b of the inverted-mesa excitation electrodes 35a and35b are curved in a convex shape.

Moreover, FIGS. 19, 20 and 21 show examples in which only one of theelectrodes in FIGS. 16, 17 and 18 has an inverted-mesa shape.

In the present invention, since the excitation electrodes of the quartzresonator are formed with an inverted-mesa shape, the capacitance ratioγ can be reduced to a value below the limit value. As a result, whenthis vibrator is used in a VCXO, the frequency variability range of theVCXO can be broadened, and the short-term stability can be improved.Moreover, when a filter is constructed, the band width is broadened, andthe impedance is lowered so that vibration is facilitated.

What is claimed is:
 1. A piezoelectric vibrator on which excitationelectrodes are respectively disposed on a top surface and undersurfaceof a piezoelectric substrate, and at least one of said excitationelectrodes is an inverted-mesa electrode comprising a projectingcircular portion and a circular recess provided in said projectingcircular portion, and wherein:a ratio of mesa thickness μ and a ratio ofmesa length are set to satisfy the equation defined as k₃ tanh k₃ (b-a)=k₂ tan k₂ (a -p), so that a capacitance ratio γ of said piezoelectricvibrator is less than 200, wherein said circular recess in saidprojecting circular portion of said inverted mesa electrode is definedas region I, said projecting circular portion surrounding said recess isdefined as region II, a region outside of said projecting circularportion of said piezoelectric substrate is defined as region III, f₁ isa cut-off frequency in said region I, f₂ is a cut-off frequency in saidregion II, f₃ is a cut-off frequency in said region III, f is aresonance frequency of said piezoelectric vibrator, k₂ is a propagationconstant for said region II and obtained by equation: ##EQU13## k₃ is apropagation constant for said region III and obtained by equation:##EQU14## p is a distance from a center of said circular recesssurrounded by said projecting circular portion to an inner edge of saidprojecting circular portion, a is a distance from said center to anouter edge of said projecting circular portion, b is a distance fromsaid center to an outer edge of said piezoelectric substrate, H is athickness of said region III, said ratio of mesa thickness μ is obtainedby equation:

    μ=(f.sub.1 -f.sub.2)/f.sub.3 -f.sub.1), and

said ratio of mesa length is equal to p/a.
 2. A piezoelectric vibratoraccording to claim 1, characterized in that said ratio of mesa thicknessμ and said ratio of mesa length are set so that said equation defined ask₃ tanh k₃ (b -a) =k₂ tan k₂ (a-p) is satisfied, and an energy trappingfactor at a time normalized resonance frequency ψ is defined as ψ=0 isless than an energy trapping factor at a time inharmonic spurious ofsaid piezoelectric vibrator is generated, whereina mass loading effectof said region I with respect to said region III is Δ, an energytrapping factor is ##EQU15## and a normalized resonance frequency ψ isobtained by equation:

    ψ=(f-f.sub.1)/(f.sub.3 -f.sub.1).


3. A piezoelectric vibrator according to claim 1, characterized in thatsaid ratio of mesa thickness μ and said ratio of mesa length are set sothat said equation defined as k₃ tanh k₃ (b-a) =k₂ tan k₂ (a -p) issatisfied, and an energy trapping factor at a time normalized resonancefrequency ψ is defined as ψ=0 is equal to an energy trapping factor at atime inharmonic spurious of said piezoelectric vibrator is generated,whereina mass loading effect of said region I with respect to saidregion III is Δ, an energy trapping factor is ##EQU16## and a normalizedresonance frequency ψ is obtained by equation:

    ψ=(f-f.sub.1)/(f.sub.3 -f.sub.1).


4. A piezoelectric vibrator according to claim 1, 2 or 3, characterizedin that said piezoelectric substrate is an AT-cut quartz crystalsubstrate.
 5. A piezoelectric vibrator according to claim 1, 2 or 3,characterized in that said piezoelectric substrate is of a shape havingsaid projecting circular portion on at least one side thereof, and saidexcitation electrode is disposed on a surface of said projectingcircular portion and on a surface surrounded by said projecting circularportion.
 6. An AT-cut quartz crystal substrate piezoelectric vibrator inwhich a piezoelectric substrate thereof is an inverted-mesa electrodethat is of a shape having a projecting circular portion on at least oneside thereof, and an excitation electrode is disposed on a surface ofsaid projecting circular portion and on a surface surrounded by saidprojecting circular portion, and wherein:a ratio of mesa thickness μ anda ratio of mesa length are set so that said equation defined as k₃ tanhk₃ (b-a) =k₂ tan k₂ (a-p) is satisfied and an energy trapping factor ata time normalized resonance frequency ψ is defined as ψ=0 is less thanan energy trapping factor at a time inharmonic spurious of saidpiezoelectric vibrator is generated, so that inharmonic spurious isrestrained from generating and a capacitance ratio γ of saidpiezoelectric vibrator is less than 200, wherein said circular recess insaid projecting circular portion of said inverted mesa electrode isdefined as region I, said projecting circular portion surrounding saidrecess is defined as region II, a region outside of said projectingcircular portion of said piezoelectric substrate is defined as regionIII, f₁ is a cut-off frequency in said region I, f₂ is a cut-offfrequency in said region II, f₃ is a cut-off frequency in said regionIII, f is a resonance frequency of said quartz crystal substrate, k₂ isa propagation constant for said region II and obtained by equation:##EQU17## k₃ is a propagation constant for said region III and obtainedby equation: ##EQU18## p is a distance from a center of said circularrecess surrounded by said projecting circular portion to an inner edgeof said projecting circular portion, a is a distance from said center toan outer edge of said projecting circular portion, b is a distance fromsaid center to an outer edge of said piezoelectric substrate, H is athickness of said region III, said ratio of mesa thickness μ is obtainedby equation:

    μ=(f.sub.1 -f.sub.2)/(f.sub.3 -f ),

said ratio of mesa length is equal to p/a, a mass loading effect of saidregion I with respect to said region III is Δ, an energy trapping factoris ##EQU19## and a normalized resonance frequency ψ is obtained byequation:

    ψ=(f-f.sub.1)/(f.sub.3 -f.sub.1).


7. An AT-cut quartz crystal substrate piezoelectric vibrator in which apiezoelectric substrate thereof is an inverted-mesa electrode that is ofa shape having a projecting circular portion on at least one sidethereof, and an excitation electrode is disposed on a surface of saidprojecting circular portion and on a surface surrounded by saidprojecting circular portion, and wherein:a ratio of mesa thickness μ anda ratio of mesa length are set so that said equation defined as k₃ tanhk₃ (b-a) k₂ tan k₂ (a-p) is satisfied and an energy trapping factor at atime normalized resonance frequency ψ is defined as ψ=0 is equal to anenergy trapping factor at a time inharmonic spurious of saidpiezoelectric vibrator is generated, so that inharmonic spurious isrestrained from generating and a capacitance ratio γ of saidpiezoelectric vibrator is less than 200, wherein said circular recess insaid projecting circular portion of said inverted mesa electrode isdefined as region I, said projecting circular portion surrounding saidrecess is defined as region II, a region outside of said projectingcircular portion of said piezoelectric substrate is defined as regionIII, f₁ is a cut-off frequency in said region I, f₂ is a cut-offfrequency in said region II, f₃ is a cut-off frequency in said regionIII, f is a resonance frequency of said quartz crystal substrate, k₂ isa propagation constant for said region II and obtained by equation:##EQU20## k₃ is a propagation constant for said region III and obtainedby equation: ##EQU21## p is a distance from a center of said circularrecess surrounded by said projecting circular portion to an inner edgeof said projecting circular portion, a is a distance from said center toan outer edge of said projecting circular portion; b is a distance fromsaid center to an outer edge of said piezoelectric substrate, H is athickness of said region III, said ratio of mesa thickness μ is obtainedby equation:

    μ=(f.sub.1 -f.sub.2)/(f.sub.3 -f.sub.1),

said ratio of mesa length is equal to p/a, a mass loading effect of saidregion I with respect to said region III is Δ, an energy trapping factoris ##EQU22## and a normalized resonance frequency ψ is obtained byequation:

    ψ=(f-f.sub.1)/(f.sub.3 -f.sub.1).


8. A piezoelectric vibrator according to claim 1, 2, 3, 6 or 7,characterized in that said projecting circular portion is formed in astairway shape.
 9. A piezoelectric vibrator according to claim 1, 2, 3,6 or 7, characterized in that said region I of said piezoelectricsubstrate is curved in a concave shape.
 10. A piezoelectric vibratoraccording to claim 1, 2, 3, 6 or 7, characterized in that said region Iof said piezoelectric substrate is curved in a convex shape.